mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)
z : ℂ
hz : ‖z‖ < 1
s : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i
k :
  Tendsto (fun n ↦ (1 - z) * ∑ j ∈ range n, (∑ k ∈ range n, f k - ∑ k ∈ range (j + 1), f k) * z ^ j) atTop
    (𝓝 (l - ∑' (i : ℕ), f i * z ^ i))
this :
  Tendsto
    (fun x ↦
      (1 - z) * ∑ i ∈ range x, (l - s x) * z ^ i +
        (1 - z) * ∑ j ∈ range x, (∑ k ∈ range x, f k - ∑ k ∈ range (j + 1), f k) * z ^ j)
    atTop (𝓝 ((1 - z) * 0 + (l - ∑' (i : ℕ), f i * z ^ i)))
n : ℕ
| (1 - z) * ∑ x ∈ range n, ((l - s n) * z ^ x + (∑ k ∈ range n, f k - ∑ k ∈ range (x + 1), f k) * z ^ x)

Tactic:
2,

Goal state after:
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)
z : ℂ
hz : ‖z‖ < 1
s : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i
k :
  Tendsto (fun n ↦ (1 - z) * ∑ j ∈ range n, (∑ k ∈ range n, f k - ∑ k ∈ range (j + 1), f k) * z ^ j) atTop
    (𝓝 (l - ∑' (i : ℕ), f i * z ^ i))
this :
  Tendsto
    (fun x ↦
      (1 - z) * ∑ i ∈ range x, (l - s x) * z ^ i +
        (1 - z) * ∑ j ∈ range x, (∑ k ∈ range x, f k - ∑ k ∈ range (j + 1), f k) * z ^ j)
    atTop (𝓝 ((1 - z) * 0 + (l - ∑' (i : ℕ), f i * z ^ i)))
n : ℕ
| ∑ x ∈ range n, ((l - s n) * z ^ x + (∑ k ∈ range n, f k - ∑ k ∈ range (x + 1), f k) * z ^ x)
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)
z : ℂ
hz : ‖z‖ < 1
s : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i
k :
  Tendsto (fun n ↦ (1 - z) * ∑ j ∈ range n, (∑ k ∈ range n, f k - ∑ k ∈ range (j + 1), f k) * z ^ j) atTop
    (𝓝 (l - ∑' (i : ℕ), f i * z ^ i))
this :
  Tendsto
    (fun x ↦
      (1 - z) * ∑ i ∈ range x, (l - s x) * z ^ i +
        (1 - z) * ∑ j ∈ range x, (∑ k ∈ range x, f k - ∑ k ∈ range (j + 1), f k) * z ^ j)
    atTop (𝓝 ((1 - z) * 0 + (l - ∑' (i : ℕ), f i * z ^ i)))
n : ℕ
| ∑ x ∈ range n, ((l - s n) * z ^ x + (∑ k ∈ range n, f k - ∑ k ∈ range (x + 1), f k) * z ^ x)

Tactic:
2,

Goal state after:
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)
z : ℂ
hz : ‖z‖ < 1
s : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i
k :
  Tendsto (fun n ↦ (1 - z) * ∑ j ∈ range n, (∑ k ∈ range n, f k - ∑ k ∈ range (j + 1), f k) * z ^ j) atTop
    (𝓝 (l - ∑' (i : ℕ), f i * z ^ i))
this :
  Tendsto
    (fun x ↦
      (1 - z) * ∑ i ∈ range x, (l - s x) * z ^ i +
        (1 - z) * ∑ j ∈ range x, (∑ k ∈ range x, f k - ∑ k ∈ range (j + 1), f k) * z ^ j)
    atTop (𝓝 ((1 - z) * 0 + (l - ∑' (i : ℕ), f i * z ^ i)))
n : ℕ
| fun x ↦ (l - s n) * z ^ x + (∑ k ∈ range n, f k - ∑ k ∈ range (x + 1), f k) * z ^ x
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: case right
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z ↦ ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
⊢ ‖(1 - z) * ∑ i ∈ range (max B₁ B₂), (l - s (i + 1)) * z ^ i‖ < ε / 2

Tactic:
calc
    _ = ‖(1 - z) * ∑ i ∈ range B₁, (l - s (i + 1)) * z ^ i +
        (1 - z) * ∑ i ∈ Ico B₁ (max B₁ B₂), (l - s (i + 1)) * z ^ i‖ := by
      rw [← mul_add, sum_range_add_sum_Ico _ (le_max_left B₁ B₂)]
    _ ≤ ‖(1 - z) * ∑ i ∈ range B₁, (l - s (i + 1)) * z ^ i‖ +
        ‖(1 - z) * ∑ i ∈ Ico B₁ (max B₁ B₂), (l - s (i + 1)) * z ^ i‖ := norm_add_le _ _
    _ = ‖1 - z‖ * ‖∑ i ∈ range B₁, (l - s (i + 1)) * z ^ i‖ +
        ‖1 - z‖ * ‖∑ i ∈ Ico B₁ (max B₁ B₂), (l - s (i + 1)) * z ^ i‖ := by
      rw [norm_mul, norm_mul]
    _ ≤ ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i +
        ‖1 - z‖ * ∑ i ∈ Ico B₁ (max B₁ B₂), ‖l - s (i + 1)‖ * ‖z‖ ^ i := by
      gcongr <;> simp_rw [← norm_pow, ← norm_mul, norm_sum_le]

Goal state after:
case right.calc.step
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z ↦ ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
⊢ ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i + ‖1 - z‖ * ∑ i ∈ Ico B₁ (max B₁ B₂), ‖l - s (i + 1)‖ * ‖z‖ ^ i <
    ε / 2
Module: Mathlib.Analysis.Convex.SpecificFunctions.Deriv

Goal state before:
pp: case hf''
m : ℤ
hm₀ : m ≠ 0
hm₁ : m ≠ 1
⊢ ∀ x ∈ Ioi 0, 0 < deriv^[2] (fun x ↦ x ^ m) x

Tactic:
intro x

Goal state after:
case hf''
m : ℤ
hm₀ : m ≠ 0
hm₁ : m ≠ 1
x : ℝ
⊢ x ∈ Ioi 0 → 0 < deriv^[2] (fun x ↦ x ^ m) x
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex

Goal state before:
pp: θ : ℂ
⊢ tan θ = 0 ↔ ∃ k, ↑k * ↑π / 2 = θ

Tactic:
rw [tan, div_eq_zero_iff, ← mul_eq_zero, ← mul_right_inj' two_ne_zero, mul_zero,
    ← mul_assoc, ← sin_two_mul, sin_eq_zero_iff]

Goal state after:
θ : ℂ
⊢ (∃ k, 2 * θ = ↑k * ↑π) ↔ ∃ k, ↑k * ↑π / 2 = θ
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf' : DifferentiableOn ℝ f (interior D)
hf'_mono : MonotoneOn (deriv f) (interior D)
⊢ ∀ {x y z : ℝ}, x ∈ D → z ∈ D → x < y → y < z → (f y - f x) / (y - x) ≤ (f z - f y) / (z - y)

Tactic:
intro x

Goal state after:
D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf' : DifferentiableOn ℝ f (interior D)
hf'_mono : MonotoneOn (deriv f) (interior D)
x : ℝ
⊢ ∀ {y z : ℝ}, x ∈ D → z ∈ D → x < y → y < z → (f y - f x) / (y - x) ≤ (f z - f y) / (z - y)
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z ↦ ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
S₁ : ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
⊢ ‖1 - z‖ * (ε / 4 / M) * (1 - ‖z‖)⁻¹ = ‖1 - z‖ * (ε / 4 / M) / (1 - ‖z‖)

Tactic:
← div_eq_mul_inv

Goal state after:
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z ↦ ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
S₁ : ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
⊢ ‖1 - z‖ * (ε / 4 / M) / (1 - ‖z‖) = ‖1 - z‖ * (ε / 4 / M) / (1 - ‖z‖)
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: case right.calc.step
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z ↦ ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
S₁ : ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
S₂ : ‖1 - z‖ * ∑ i ∈ Ico B₁ (max B₁ B₂), ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
⊢ ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i + ‖1 - z‖ * ∑ i ∈ Ico B₁ (max B₁ B₂), ‖l - s (i + 1)‖ * ‖z‖ ^ i <
    ε / 2

Tactic:
convert! add_lt_add S₁ S₂ using 1

Goal state after:
case e'_4
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z ↦ ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
S₁ : ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
S₂ : ‖1 - z‖ * ∑ i ∈ Ico B₁ (max B₁ B₂), ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
⊢ ε / 2 = ε / 4 + ε / 4
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex

Goal state before:
pp: ⊢ Function.Surjective cos

Tactic:
intro x

Goal state after:
x : ℂ
⊢ ∃ a, cos a = x
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex

Goal state before:
pp: ⊢ Function.Surjective sin

Tactic:
intro x

Goal state after:
x : ℂ
⊢ ∃ a, sin a = x
Module: Mathlib.Analysis.Complex.AbsMax

Goal state before:
pp: case inr
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℂ E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : Nontrivial E
f : E → F
U : Set E
hU : Bornology.IsBounded U
hd : DiffContOnCl ℂ f U
C : ℝ
hC : ∀ z ∈ frontier U, ‖f z‖ ≤ C
z : E
hz : z ∈ U
w : E
hne : w ≠ z
e : ℂ → E := ⇑(lineMap z w)
hde : Differentiable ℂ e
hL : AntilipschitzWith (nndist z w)⁻¹ e
⊢ ‖f z‖ ≤ C

Tactic:
replace hd : DiffContOnCl ℂ (f ∘ e) (e ⁻¹' U) :=
    hd.comp hde.diffContOnCl (mapsTo_preimage _ _)

Goal state after:
case inr
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℂ E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : Nontrivial E
f : E → F
U : Set E
hU : Bornology.IsBounded U
C : ℝ
hC : ∀ z ∈ frontier U, ‖f z‖ ≤ C
z : E
hz : z ∈ U
w : E
hne : w ≠ z
e : ℂ → E := ⇑(lineMap z w)
hde : Differentiable ℂ e
hL : AntilipschitzWith (nndist z w)⁻¹ e
hd : DiffContOnCl ℂ (f ∘ e) (e ⁻¹' U)
⊢ ‖f z‖ ≤ C
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex

Goal state before:
pp: case mpr.inl
x : ℝ
n : ℤ
h : ↑(n + n) * π = x
⊢ (∃ n, ↑n * (2 * π) = x) ∨ ∃ k, π + ↑k * (2 * π) = x

Tactic:
exact .inl ⟨n, by grind⟩

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: x : ℝ
⊢ tan (2 * x) = 2 * tan x / (1 - tan x ^ 2)

Tactic:
have := @Complex.tan_two_mul x

Goal state after:
x : ℝ
this : Complex.tan (2 * ↑x) = 2 * Complex.tan ↑x / (1 - Complex.tan ↑x ^ 2)
⊢ tan (2 * x) = 2 * tan x / (1 - tan x ^ 2)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex

Goal state before:
pp: case mpr.inl
x : ℝ
n : ℤ
h : π / 2 + ↑(n + n) * π = x
⊢ (∃ k, π / 2 + ↑k * (2 * π) = x) ∨ ∃ k, -(π / 2) + ↑k * (2 * π) = x

Tactic:
exact .inl ⟨n, by grind⟩

Goal state after:
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: S : Set ℝ
f : ℝ → ℝ
hfc : ConvexOn ℝ S f
⊢ MonotoneOn (fun x ↦ derivWithin f (Ioi x) x) (interior S)

Tactic:
intro x

Goal state after:
S : Set ℝ
f : ℝ → ℝ
hfc : ConvexOn ℝ S f
x : ℝ
⊢ x ∈ interior S →
    ∀ ⦃b : ℝ⦄, b ∈ interior S → x ≤ b → (fun x ↦ derivWithin f (Ioi x) x) x ≤ (fun x ↦ derivWithin f (Ioi x) x) b
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: case inr.refine_1
S : Set ℝ
f : ℝ → ℝ
hfc : ConvexOn ℝ S f
x : ℝ
hxs : x ∈ interior S
y : ℝ
hys : y ∈ interior S
hxy✝ : x ≤ y
hxy : x < y
hys' : ∃ l u, y ∈ Ioo l u ∧ Ioo l u ⊆ S
⊢ (slope f y '' {y_1 | y_1 ∈ S ∧ y < y_1}).Nonempty

Tactic:
obtain ⟨a, b, hxab, habs⟩ := hys'

Goal state after:
case inr.refine_1
S : Set ℝ
f : ℝ → ℝ
hfc : ConvexOn ℝ S f
x : ℝ
hxs : x ∈ interior S
y : ℝ
hys : y ∈ interior S
hxy✝ : x ≤ y
hxy : x < y
a b : ℝ
hxab : y ∈ Ioo a b
habs : Ioo a b ⊆ S
⊢ (slope f y '' {y_1 | y_1 ∈ S ∧ y < y_1}).Nonempty
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: S : Set ℝ
f : ℝ → ℝ
hfc : ConvexOn ℝ S f
⊢ MonotoneOn (fun x ↦ derivWithin f (Iio x) x) (interior S)

Tactic:
intro x

Goal state after:
S : Set ℝ
f : ℝ → ℝ
hfc : ConvexOn ℝ S f
x : ℝ
⊢ x ∈ interior S →
    ∀ ⦃b : ℝ⦄, b ∈ interior S → x ≤ b → (fun x ↦ derivWithin f (Iio x) x) x ≤ (fun x ↦ derivWithin f (Iio x) x) b
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
⊢ sin (arccos (⟪x, y⟫ / (‖x‖ * ‖y‖))) * (‖x‖ * ‖y‖) = √(⟪x, x⟫ * ⟪y, y⟫ - ⟪x, y⟫ * ⟪x, y⟫)

Tactic:
rw [Real.sin_arccos]

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
⊢ √(1 - (⟪x, y⟫ / (‖x‖ * ‖y‖)) ^ 2) * (‖x‖ * ‖y‖) = √(⟪x, x⟫ * ⟪y, y⟫ - ⟪x, y⟫ * ⟪x, y⟫)
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: S : Set ℝ
f : ℝ → ℝ
hfc : ConvexOn ℝ S f
hfd : DifferentiableOn ℝ f S
⊢ MonotoneOn (derivWithin f S) S

Tactic:
intro x

Goal state after:
S : Set ℝ
f : ℝ → ℝ
hfc : ConvexOn ℝ S f
hfd : DifferentiableOn ℝ f S
x : ℝ
⊢ x ∈ S → ∀ ⦃b : ℝ⦄, b ∈ S → x ≤ b → derivWithin f S x ≤ derivWithin f S b
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: S : Set ℝ
f : ℝ → ℝ
hfc : ConvexOn ℝ S f
hfd : ∀ x ∈ S, DifferentiableAt ℝ f x
⊢ MonotoneOn (deriv f) S

Tactic:
intro x

Goal state after:
S : Set ℝ
f : ℝ → ℝ
hfc : ConvexOn ℝ S f
hfd : ∀ x ∈ S, DifferentiableAt ℝ f x
x : ℝ
⊢ x ∈ S → ∀ ⦃b : ℝ⦄, b ∈ S → x ≤ b → deriv f x ≤ deriv f b
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: S : Set ℝ
f : ℝ → ℝ
hfc : StrictConvexOn ℝ S f
hfd : DifferentiableOn ℝ f S
⊢ StrictMonoOn (derivWithin f S) S

Tactic:
intro x

Goal state after:
S : Set ℝ
f : ℝ → ℝ
hfc : StrictConvexOn ℝ S f
hfd : DifferentiableOn ℝ f S
x : ℝ
⊢ x ∈ S → ∀ ⦃b : ℝ⦄, b ∈ S → x < b → derivWithin f S x < derivWithin f S b
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: S : Set ℝ
f : ℝ → ℝ
hfc : StrictConvexOn ℝ S f
hfd : ∀ x ∈ S, DifferentiableAt ℝ f x
⊢ StrictMonoOn (deriv f) S

Tactic:
intro x

Goal state after:
S : Set ℝ
f : ℝ → ℝ
hfc : StrictConvexOn ℝ S f
hfd : ∀ x ∈ S, DifferentiableAt ℝ f x
x : ℝ
⊢ x ∈ S → ∀ ⦃b : ℝ⦄, b ∈ S → x < b → deriv f x < deriv f b
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: S : Set ℝ
f : ℝ → ℝ
hfc : ConcaveOn ℝ S f
hfd : DifferentiableOn ℝ f S
⊢ AntitoneOn (derivWithin f S) S

Tactic:
intro x

Goal state after:
S : Set ℝ
f : ℝ → ℝ
hfc : ConcaveOn ℝ S f
hfd : DifferentiableOn ℝ f S
x : ℝ
⊢ x ∈ S → ∀ ⦃b : ℝ⦄, b ∈ S → x ≤ b → derivWithin f S b ≤ derivWithin f S x
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: S : Set ℝ
f : ℝ → ℝ
hfc : StrictConcaveOn ℝ S f
hfd : DifferentiableOn ℝ f S
⊢ StrictAntiOn (derivWithin f S) S

Tactic:
intro x

Goal state after:
S : Set ℝ
f : ℝ → ℝ
hfc : StrictConcaveOn ℝ S f
hfd : DifferentiableOn ℝ f S
x : ℝ
⊢ x ∈ S → ∀ ⦃b : ℝ⦄, b ∈ S → x < b → derivWithin f S b < derivWithin f S x
Module: Mathlib.Analysis.Complex.BorelCaratheodory

Goal state before:
pp: f : ℂ → ℂ
M R : ℝ
z : ℂ
hM : 0 < M
hf : DifferentiableOn ℂ f (ball 0 R)
hf₁ : Set.MapsTo f (ball 0 R) {z | z.re ≤ M}
hR : 0 < R
hz : z ∈ ball 0 R
⊢ Set.MapsTo (fun {z} ↦ f z - f 0) (ball 0 R) {z | z.re ≤ M + ‖f 0‖}

Tactic:
intro x

Goal state after:
f : ℂ → ℂ
M R : ℝ
z : ℂ
hM : 0 < M
hf : DifferentiableOn ℂ f (ball 0 R)
hf₁ : Set.MapsTo f (ball 0 R) {z | z.re ≤ M}
hR : 0 < R
hz : z ∈ ball 0 R
x : ℂ
⊢ x ∈ ball 0 R → (fun {z} ↦ f z - f 0) ∈ {z | z.re ≤ M + ‖f 0‖}
Module: Mathlib.Analysis.Complex.BranchLogRoot

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : LocallyPathConnectedSpace X
U : Set X
hUc : IsSimplyConnected U
hUo : IsOpen U
g✝ : X → ℂ
hgc : ContinuousOn g✝ U
hU₀ : 0 ∉ g✝ '' U
this✝ : SimplyConnectedSpace ↑U
this : LocallyPathConnectedSpace ↑U
x₀ : { x // x ∈ U }
hx₀ : g✝ ↑x₀ ≠ 0
f : C(↑U, ℂ)
hf : cexp ∘ ⇑f = ⇑{ toFun := U.domRestrict g✝, continuous_toFun := ⋯ }
g : X → ℂ
hg : ∀ (z : ↑U), g ↑z = f z
⊢ EqOn (cexp ∘ g) g✝ U

Tactic:
intro x

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : LocallyPathConnectedSpace X
U : Set X
hUc : IsSimplyConnected U
hUo : IsOpen U
g✝ : X → ℂ
hgc : ContinuousOn g✝ U
hU₀ : 0 ∉ g✝ '' U
this✝ : SimplyConnectedSpace ↑U
this : LocallyPathConnectedSpace ↑U
x₀ : { x // x ∈ U }
hx₀ : g✝ ↑x₀ ≠ 0
f : C(↑U, ℂ)
hf : cexp ∘ ⇑f = ⇑{ toFun := U.domRestrict g✝, continuous_toFun := ⋯ }
g : X → ℂ
hg : ∀ (z : ↑U), g ↑z = f z
x : X
⊢ x ∈ U → (cexp ∘ g) x = g✝ x
Module: Mathlib.Analysis.Complex.BranchLogRoot

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : LocallyPathConnectedSpace X
U : Set X
hUc : IsSimplyConnected U
hUo : IsOpen U
g : X → 𝔻
hgc : ContinuousOn g U
hU₀ : 0 ∉ g '' U
n : ℕ+
f : X → ℂ
hfc : ContinuousOn f U
hf : ∀ (x : X), f x ^ ↑n = (UnitDisc.coe ∘ g) x
⊢ ∀ (x : X), ‖f x‖ < 1

Tactic:
intro x

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : LocallyPathConnectedSpace X
U : Set X
hUc : IsSimplyConnected U
hUo : IsOpen U
g : X → 𝔻
hgc : ContinuousOn g U
hU₀ : 0 ∉ g '' U
n : ℕ+
f : X → ℂ
hfc : ContinuousOn f U
hf : ∀ (x : X), f x ^ ↑n = (UnitDisc.coe ∘ g) x
x : X
⊢ ‖f x‖ < 1
Module: Mathlib.Analysis.Meromorphic.IsolatedZeros

Goal state before:
pp: case h.right
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f g : 𝕜 → E
hf : MeromorphicOn f U
hg : MeromorphicOn g U
hU : Preperfect U
h : f =ᶠ[codiscreteWithin U] g
⊢ U ⊆ {x | f =ᶠ[𝓝[≠] x] g}

Tactic:
intro x

Goal state after:
case h.right
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f g : 𝕜 → E
hf : MeromorphicOn f U
hg : MeromorphicOn g U
hU : Preperfect U
h : f =ᶠ[codiscreteWithin U] g
x : 𝕜
⊢ x ∈ U → x ∈ {x | f =ᶠ[𝓝[≠] x] g}
Module: Mathlib.Analysis.Meromorphic.IsolatedZeros

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f g : 𝕜 → E
hf : MeromorphicOn f U
hg : MeromorphicOn g U
h : f =ᶠ[codiscreteWithin U] g
⊢ ∀ x ∈ U, {x | deriv f x = deriv g x} ∪ Uᶜ ∈ 𝓝[≠] x

Tactic:
intro x

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f g : 𝕜 → E
hf : MeromorphicOn f U
hg : MeromorphicOn g U
h : f =ᶠ[codiscreteWithin U] g
x : 𝕜
⊢ x ∈ U → {x | deriv f x = deriv g x} ∪ Uᶜ ∈ 𝓝[≠] x
Module: Mathlib.Analysis.Meromorphic.Divisor

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
U : Set 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
hf : AnalyticOnNhd 𝕜 f U
⊢ 0 ≤ divisor f U

Tactic:
intro x

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
U : Set 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
hf : AnalyticOnNhd 𝕜 f U
x : 𝕜
⊢ 0 x ≤ (divisor f U) x
Module: Mathlib.Analysis.Meromorphic.TrailingCoefficient

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
h : meromorphicOrderAt f x = ⊤
⊢ meromorphicTrailingCoeffAt f x = 0

Tactic:
simp_all [meromorphicTrailingCoeffAt]

Goal state after:
Module: Mathlib.Analysis.Meromorphic.NormalForm

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : 𝕜
f g : 𝕜 → E
hfg : f =ᶠ[𝓝[≠] x] g
⊢ toMeromorphicNFAt f x =ᶠ[𝓝 x] toMeromorphicNFAt g x

Tactic:
by_cases hf : MeromorphicAt f x
  · exact meromorphicNFAt_toMeromorphicNFAt.eventuallyEq_nhdsNE_iff_eventuallyEq_nhds
        meromorphicNFAt_toMeromorphicNFAt |>.mp <| hf.eq_nhdsNE_toMeromorphicNFAt.symm.trans
      <| hfg.trans ((MeromorphicAt.meromorphicAt_congr hfg).mp hf).eq_nhdsNE_toMeromorphicNFAt
  · simp [hf, MeromorphicAt.meromorphicAt_congr hfg |>.not.mp]

Goal state after:
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f : 𝕜 → E
hf : MeromorphicOn f U
h'f : ∀ x ∈ U, meromorphicOrderAt f x ≠ ⊤
⊢ ∀ x ∈ U, (U \ {x | f x ≠ 0})ᶜ ∈ 𝓝[≠] x

Tactic:
intro x

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f : 𝕜 → E
hf : MeromorphicOn f U
h'f : ∀ x ∈ U, meromorphicOrderAt f x ≠ ⊤
x : 𝕜
⊢ x ∈ U → (U \ {x | f x ≠ 0})ᶜ ∈ 𝓝[≠] x
Module: Mathlib.Analysis.Meromorphic.NormalForm

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
U : Set 𝕜
hf : MeromorphicOn f U
⊢ EqOn (toMeromorphicNFOn f U) f Uᶜ

Tactic:
intro x

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
U : Set 𝕜
hf : MeromorphicOn f U
x : 𝕜
⊢ x ∈ Uᶜ → toMeromorphicNFOn f U x = f x
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: case inl
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
hf : MeromorphicAt f x
ho : meromorphicOrderAt f x = 0
⊢ (∃ c, c ≠ 0 ∧ Tendsto f (𝓝[≠] x) (𝓝 c)) ↔ meromorphicOrderAt f x = 0

Tactic:
simp [ho, tendsto_ne_zero_of_meromorphicOrderAt_eq_zero hf ho]

Goal state after:
Module: Mathlib.Analysis.Meromorphic.FactorizedRational

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
u₀ : 𝕜
hd : HasFiniteSupport d
h₁d : ¬d u₀ = 0
this✝ : (Function.mulSupport fun u ↦ (fun x ↦ x - u) ^ d u) ⊆ ↑(Finite.toFinset hd)
this : u₀ ∈ Finite.toFinset hd
⊢ support (update d u₀ 0) ⊆ ↑((Finite.toFinset hd).erase u₀)

Tactic:
intro x

Goal state after:
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
u₀ : 𝕜
hd : HasFiniteSupport d
h₁d : ¬d u₀ = 0
this✝ : (Function.mulSupport fun u ↦ (fun x ↦ x - u) ^ d u) ⊆ ↑(Finite.toFinset hd)
this : u₀ ∈ Finite.toFinset hd
x : 𝕜
⊢ x ∈ support (update d u₀ 0) → x ∈ ↑((Finite.toFinset hd).erase u₀)
Module: Mathlib.Analysis.Meromorphic.FactorizedRational

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
hd : ¬(Function.mulSupport fun u ↦ (fun x ↦ x - u) ^ d u).Finite
⊢ MeromorphicNFOn (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ d u) univ

Tactic:
rw [finprod_of_infinite_mulSupport hd]

Goal state after:
case neg
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
hd : ¬(Function.mulSupport fun u ↦ (fun x ↦ x - u) ^ d u).Finite
⊢ MeromorphicNFOn 1 univ
Module: Mathlib.Analysis.Meromorphic.FactorizedRational

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
x : 𝕜
h : (support d).Finite
⊢ (Function.mulSupport fun u ↦ (x - u) ^ update d x 0 u) ⊆ ↑h.toFinset

Tactic:
by
  intro u
  contrapose
  simp only [mem_mulSupport, ne_eq, Decidable.not_not]
  by_cases h₁ : u = x
  · rw [h₁]
    simp
  · simp_all

Goal state after:
Module: Mathlib.Analysis.Complex.CanonicalDecomposition

Goal state before:
pp: R : ℝ
w : ℂ
⊢ Meromorphic (canonicalFactor R w)

Tactic:
intro x

Goal state after:
R : ℝ
w x : ℂ
⊢ MeromorphicAt (canonicalFactor R w) x
Module: Mathlib.Analysis.Complex.CanonicalDecomposition

Goal state before:
pp: R : ℝ
w : ℂ
⊢ AnalyticOnNhd ℂ (canonicalFactor R w) {w}ᶜ

Tactic:
intro x

Goal state after:
R : ℝ
w x : ℂ
⊢ x ∈ {w}ᶜ → AnalyticAt ℂ (canonicalFactor R w) x